The finite element method (FEM) (sometimes referred to as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge-Kutta, etc.
In simulating structural mechanics, an engineering structure or product (e.g., car, cellular phone, airplane, etc.) can be modeled with a set of finite elements interconnected through nodal points or nodes. Each finite element is configured to have a shape and physical properties such as density, Young's modulus, shear modulus and Possion's ratio, and alike. Finite element can be one-, two- or three-dimensional. In general, a three-dimensional element is referred to as a solid element (i.e., a finite element having a volume). One of the most common solid elements is 8-node hexahedral element 100 or brick element shown in FIG. 1A. Eight-node hexahedral element 100 is a first order finite element that contains eight corner nodes. Shown in FIG. 1B is a two-dimensional view from one side of the 8-node hexahedral element of FIG. 1A.
To evaluate finite element results (e.g., nodal forces being generated by stress within an element), each hexahedral element is configured with one or more integration points for numerical integration, for example, Gauss-Legendre quadrature numerical integration scheme. Numerical integration of a hexahedral element can be done with a single Gauss-Legendre integration point. Such element is referred to as an under-integrated or rank deficient element (not shown). Alternatively, a hexahedral element 100 uses two Gauss-Legendre integration points 102 in each spatial direction for a total of eight points. Such element is said to have full integration or rank sufficient integration. Full integration guarantees that all possible modes of deformation generate stress in the element.
Further, finite element method uses a set of shape functions Ni for each element to construct approximated displacement uh anywhere within the element in accordance with the following formula:
      u    h    =            ∑      i        ⁢                  N        i            ⁢              u        i            where ui, is the nodal displacement. Each node has three translational displacements, therefore, i is 24 for an 8-node hexahedral element (i.e., 8 nodes with each having three displacements).
A fully integrated 8-node hexahedral element suffers from what is referred to as shear locking effect, which means that a built-in artificial shear stiffness for certain deformation modes due to the placement of the integration points. This spurious stiffness is even more prominent for elements with poor aspect ratio, i.e., for element with one of the spatial dimensions substantially larger than another. For example, shown in FIG. 2A, an elongated hexahedral element 200 is said to have a poor aspect ratio (i.e., substantially deviated from 1). To better view the relationship between the integration points 202 and the element 200, a two-dimensional view of the element is shown in FIG. 2B. The aspect ratio is defined as ratio of respective lengths of two sides, W 212 and H 214 in FIG. 2B. In three-dimension, there are three aspect ratios one for each spatial dimension.
Sometimes it is more advantageous to create a finite element analysis model with solid elements with poor aspect ratio due to geometry of an engineering product or structure (e.g., a thin-walled structure). The advantage includes at least the following: 1) easier to create the model; and 2) more computational efficient due to less number of elements in the model.
Generally, a fully-integrated 8-node solid element has a numerical deficiency referred to as transverse shear locking in simulating pure bending. And the shear locking effect is amplified when the solid elements have poor aspect ratio. FIG. 3A is a two-dimensional side view showing shear locking effect of an 8-node solid element. Diagram 310 of FIG. 3A shows a realistic pure bending of a prism or elongated structure, while diagram 320 of FIG. 3B shows a poor aspect ratio 8-node solid element under the same bending moment 300. It is evident that 8-node hexahedral element presents no curvature between nodes; therefore, the 8-node hexahedral element is numerically too stiff in comparison to the true structural behaviors it supposed to simulate. For fully-integrated 8-node solid element, integration points (202 of FIG. 2A) are not located in the centroid of the solid element and whence this shear locking effect.
One prior art approach to solve this shear locking problem is to use higher order elements, for example, 20-node element (one additional node per edge, not shown). However, the computation costs associated with the higher order elements prevent practical usage in any real world production situations. It would, therefore, be desirable to provide an improved 8-node hexahedral element configured for reducing shear-locking in finite element method.